setproofskey - McCombs Math 381 Proofs and Set Operations...

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McCombs Math 381 Proofs and Set Operations Given finite sets A and B . 1. A ! B = x | x " A or x " B { } 2. A ! B = x | x " A and x " B { } 3. A ! B = x , y ( ) | x " A , y " B { } 4. A ! B = x | x " A and x # B { } 5. A ! B = A " B ( ) # B " A ( ) 6. P A ( ) = S | S ! A { } 7. A = x | x ! A { } 8. P A ( ) = 2 A 9. A ! B = A + B " A # B 10. A ! B = A " B 11. A ! B " C ( ) = A ! B ( ) # A ! C ( ) 12. A ! B " C ( ) = A ! B ( ) # A ! C ( ) Proof Strategies 1. To prove A ! B , we show x ! A " x ! B . 2. To prove A = B , we show A ! B and B ! A . Examples: Prove each statement. 1. Given set A , !" A . Proof: We need to show that x !"# x ! A . This statement is vacuously true since x !" is a FALSE statement. 2. There is only one empty set. Proof: Suppose we have two sets E 1 and E 2 each of which is empty. By Example 1 above, E 1 empty implies E 1 ! E 2 . Similarly, E 2 empty implies E 2 ! E 1 . Thus, E 1 = E 2 . Therefore, there is only one empty set. 3. Given integers c and d , let C = x ! Z : x c { } , and D = x ! Z : x d { } . Prove that C ! D if and only if c d . Proof: Part 1: C ! D " c d . Assume C ! D . This means that every element in C is also an element in D . Note that the integer c ! C since c c .
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setproofskey - McCombs Math 381 Proofs and Set Operations...

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